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ON PROCONGRUENCE CURVE COMPLEXES AND
THEIR AUTOMORPHISMS
Pierre Lochak
To cite this version:
Pierre Lochak. ON PROCONGRUENCE CURVE COMPLEXES AND THEIR AUTOMORPHISMS.
2020. �hal-02992317�
ON PROCONGRUENCE CURVE COMPLEXES AND THEIR AUTOMORPHISMS
PIERRE LOCHAK
Abstract. In this paper we start exploring the procongruence completions of three varieties of curve complexes attached to hyperbolic surfaces, as well as their automorphisms groups. The discrete counterparts of these objects, especially the curve complex and the so-called pants complex were defined long ago and have been the subject of numerous studies. Introducing some form of completions is natural and indeed necessary to lay the ground for a topological version of Grothendieck-Teichm¨uller theory. Here we state and prove several results of fundational nature, among which reconstruction theorems in the discrete and complete settings, which give a graph theoretic characterizations of versions of the curve complex as well as a rigidity theorem for the complete pants complex, in sharp contrast with the case of the (complete) curve complex, whose automorphisms actually define a version of the Grothendieck-Teichm¨uller group, to be studied elsewhere (see [20]). We work all along with the procongruence completions – and for good reasons – recalling however that the so-called congruence conjecture predicts that this completion should coincide with the full profinite completion.
AMS Math Classification: 11R32, 14D22, 57M99.
1. Introduction
The primary goal of this paper is to start laying the foundations for a topological version of Grothendieck-Teichm¨uller theory and the goal of this short introduction is to provide some clues as to what this could mean ; and of course about the contents of the paper. For much more on the background landscape we refer once and for all to [21] and its references. Because numerous objects are involved we have gathered the main (essentially classical) definitions in a short Appendix which the reader is invited to consult when (s)he feels like it. We will also explicitly refer to it.
In a few words which will be considerably expanded below and possibly elsewhere, the situation can be described as follows. Let S = Sg,nbe a hyperbolic surface of finite type (cf. §A.1); it has (modular) dimension
d(S) = 3g − 3 + n which can be seen for instance as the (complex) dimension of the modular orbifold M(S) (cf. §A.2) or else as the maximal number of non intersecting simple closed curves lying on S, considered up to isotopy (these objects form a set which we denote L(S)). Starting from L(S) one builds several (simplicial, non locally finite) complexes, especially the curve complex C(S) (cf.§A.5), of dimension d(S) − 1, and the so-called two-dimensional pants complex CP(S) (cf. §A.7) of which it is enough to consider the
1-skeleton (the pants graph). The attached Teichm¨uller group (a.k.a. mapping class group) Γ(S) (cf. §A.3) acts naturally on these objects (L(S), C(S), CP(S)).
The curve complex C(S) was first constructed by W.J.Harvey in close analogy with buildings for reductive groups, from which the significance of its automorphisms was immediately recognized (see [21], Introduction, for a more detailed story and references). It was shown in the eighties, by N.V.Ivanov (cf. [15]) and J.L.Harer (cf. [12, 13]) independently, that the curve complex C(S) has the homotopy type of a wedge of spheres, an important and fundationnal result. A few years later N.V.Ivanov proved (cf. [16] as well as [18]) that C(S) is essentially rigid, the only automorphism not arising from the action of Γ(S) being the mirror reflection (an orientation reversing automorphism of the underlying surface). This is embodied in the exact sequence (A 2) of §A.12. An important point is that it also enables one to control the automorphisms of the group Γ(S), leading to the exact sequence (A 3), and indeed the automorphisms of any cofinite subgroup Γλ(S) ⊂ Γ(S). The upshot is thus that both C(S) and Γ(S) are rigid with the mirror reflection as only non
inner automorphism; in anticipation one can identify the reflection with complex conjugacy and consider that it generates the Galois group Gal(C/R) ' Z/2.
The pants complex CP(S) was defined somewhat later and its automorphisms were considered relatively
recently. D.Margalit showed (in [24]) that it is rigid as well, more precisely that one can replace C(S) by CP(S) in the sequence (A 2), so that Aut(CP(S)) = Aut(C(S)). This result will be reproved below (in §2)
in a different way.
Now to completions; they were introduced in [3] in an effort to attack the congruence conjecture (cf. §A.10). Although this was actually not achieved in [3] (see the review of D.Abramovitch in MathSciNet for a care-ful and well-intended discussion), the idea of completing various geometric or in fact topological objects (cf. §A.11), primarily versions of the complexes of curves, appears as a deep and potentially fruitful one. Perhaps the main point or slogan of the present paper is that the automorphisms of the completed complexes have a lot to do with Grothendieck-Teichm¨uller theory (in all genera, not only genus 0) and the corresponding group. This is also the main theme of the manuscript [20] (2007, unpublished).
More specifically let ˆC(S) and ˆCP(S) denote the respective profinite completions of the curves and pants
complexes. Then ˆCP(S) remains rigid whereas ˆC(S) acquires an enormous automorphism group, which
is precisely (a somewhat sophisticated version of) the Grothendieck-Teichm¨uller group. These issues are discussed in detail in [20] but watertight proofs are missing there, for technical reasons which in some sense amount to the fact that one does not know how to prove (the highly plausible fact) that ˆC(S) is isomorphic to the profinite completion ˆCG(S) of the group theoretic version CG(S) of the curve complex (cf. A.6).
Fortunately things become somewhat easier when working with the congruence completions. In terms of covers the congruence completion ˇΓ(S) describes the (orbifold unramified finite) covers of the modular orbifold M(S) arising from covers of S itself, which are obviously much more manageable. Whether these covers are cofinal or not is the question which the congruence conjecture purports to answer in a positive way. In any event here we take advantage of the results shown in particular in [4] to attack the questions in the framework of the congruence completions. Turning to the procongruence complexes ˇC(S) and ˇCP(S)
we prove that the latter one, namely the procongruence pants complex, remains rigid. That is we have a short exact sequence:
1 → Inn(ˇΓ(S)) → Aut(( ˇCP(S))st) → Z/2 → 1.
This is a somewhat cryptic and incomplete version of the result. The subscript st stand for “stack” and the exact definition of these objects, which involves the so-called topological stacks, will be detailed in due time, at the beginning of section 7. One can again state that (with the mild exception of type (1, 2)) Out(( ˇCP(S))st) ' Z/2 ' Gal(C/R), just as in the discrete case, and the nontrivial outer automorphism
comes again from orientation or complex conjugacy.
At first sight this may appear as a rather dull result: the procongruence pants complex is rigid, and this is also the case in the full profinite setting, modulo the congruence conjecture. In other words, rigidity survives completion in that case. So what? The point is that there are at least one surprise and one application in store. The surprise – if any – consists in the fact that the procongruence curve complex is not rigid. Far from it; indeed the outer automorphism group Out( ˇC(S)) (for d(S) > 3, say) is enormous and can be taken as a higher genus version of the Grothendieck-Teichm¨uller group. In particular it is independent of S, that is of the type (g, n), and it naturally contains the absolute Galois group Gal( ¯Q/Q). This and much more is elaborated in [20] (see also [21]) which however again does not contain watertight proofs inasmuch as the setting there is that of full profinite completion where certain tools are still lacking, in contrast with the case of the procongruence completion. The upshot is that the rigidity result shown in the present paper should pave the way for a thorough investigation of this new, topological version of Grothendieck-Teichm¨uller theory.
Acknowledgments. This paper benefited from conversations and exchanges with M. Boggi and L. Funar, who after the near completion of the present text decided to produce a different exposition of part of this material. The attentive reader will find that the differences actually overshadow some obvious similarities.
2. Discrete complexes : rigidity and reconstruction
In this section we prepare the ground by recalling some rigidity results in the discrete setting in a fashion taylored to our needs (see §A.12 for a tightly compressed summary) and prove a reconstruction result which later on will be adapted to the procongruence setting; as a side benefit it provides another proof of the main result of [24], that is the rigidity of the discrete pants complex. To a hyperbolic surface S we associate in particular three graphs, namely the 1-skeleton C(1)(S) of the curve complex (cf. §A.5), the pants graph
CP(1)(S) (cf. §A.7) and the graph C∗(S) (cf. §A.8). The definitions readily extend (cf. §A.9) to the case of not
necessarily connected surfaces, with hyperbolic connected. These three graphs, and later on their respective completions, carry all the information we need. In some sense we are trying to pass from an essentially group theoretic framework, revolving around the Teichm¨uller group Γ(S) (cf. §A.3), its completions and their cofinite subgroups to a graph theoretic one, based on the graphs above and later their completions, together with certain subgraphs.
2.1. Rigidity of the discrete curves complex. Basically this paragraph revolves around the two short exact sequences of §A.12. We start with the curve complex C(S) and consider its group of simplicial auto-morphisms Aut(C(S)). There is a natural map M od(S) → Aut(C(S)) induced by letting a diffeomorphism act on loops (i.e. elements of L(S) = C(0)(S); cf. §A.5), everything up to isotopy. The elements of the center of the left-hand group lie in the kernel of that map because they commute with twists, so there is an induced map θ : Inn(M od(S))) → Aut(C(S)). Assume now that C(S) is connected, that is d(S) > 1. Then it is not too difficult to show that θ is injective. The deep fundamental fact mentioned in the introduction and embodied by the sequence (A 2) states that θ is also surjective for (g, n) 6= (1, 2). This surjectivity, in item i) below, is due to N.V.Ivanov ([16]) and F.Luo ([18]):
Theorem 2.1. Let S be a connected hyperbolic surface of type (g, n) with d(S) > 1. Then
i) the natural map θ : Inn(M od(S)) → Aut(C(S)) is an isomorphism except if (g, n) = (1, 2), in which case it is injective but not surjective; in fact θ maps Inn(M od(S1,2)) onto the strict subgroup of the elements
Aut(C(S1,2)) which globally preserve the set of vertices representing nonseparating curves;
ii) Aut(C(1)(S)) = Aut(C(S)).
Of course, if the type is different from (1, 2) and (2, 0), M od(S) is centerfree and θ provides an isomorphism between M od(S) and Aut(C(S)). Item ii) is easy but quite telling; it confirms that the pants complex and the pants graph (i.e. its 1-skeleton) have the same automorphisms. This fact will remain valid after completion. Here is a short proof. There is a natural map Aut(C(S)) → Aut(C(1)(S)) which is injective;
indeed the restriction to the set of vertices is already injective. To prove surjectivity it is enough to give a graph theoretic characterization of the higher dimensional simplices of C(S) and this is easily available: a moment contemplation will confirm that the k-dimensional simplices are in one-to-one correspondence with the complete subgraphs (a.k.a. cliques) of C(1)(S) with k + 1 vertices, i.e. subgraphs such that any two vertices are connected by an edge. This characterization proves ii). Note that to any simplicial complex one can associate the complex obtained by adding in all the cliques as simplices. Here C(S) is a flag complex, that is, its simplices are exactly given by the cliques. This will also be the case of the other complexes we will meet (including in the profinite world) and it says that in fact all the information is contained is contained in a graph, namely the 1-skeleton of the relevant complex.
Remark 2.1. The odd looking case of type (1, 2) is actually easy to understand. It stems from the fact that C(S1,2) and C(S0,5) are isomorphic, whereas Γ1,[2]/Z(Γ1,[2]) maps into Γ0,[5] as a subgroup of index 5; indeed
θ maps Inn(M od(S1,2)) injectively onto an index 5 subgroup of Aut(C(S1,2)). See §A.4 and [20, 24] for a
geometric discussion.
N.V.Ivanov went on to show how to use the description of Aut(C(S)) afforded by Theorem 2.1 in order to study the action of Γ(S) on Teichm¨uller space. He recovered in this way ([16]) the classical result of H.Royden about automorphisms of Teichm¨uller spaces:
Corollary 2.2. If d(S) > 1, any complex automorphism of T (S) is induced by an element of M od(S). As N.V.Ivanov again showed, Theorem 2.1 also has immediate bearing on the automorphisms of the modular groups. Here we require one more definition, which will turn out to be of typical anabelian flavor: Definition 2.3. An element of Aut(Γ(S)) is called inertia preserving if it (globally) preserves the set of cyclic subgroups generated by Dehn twists, that is maps a twist in Γ(S) to a power of some other twist.
For a geometric discussion justifying this terminology we refer e.g. to [21]. In the present discrete setting we have the following
Theorem 2.4. If d(S) > 1, all automorphisms of Γ(S) are inertia preserving: Aut∗(Γ(S)) = Aut(Γ(S)). This result, which again is essentially due to N.V.Ivanov (cf. [15] and references therein) rests on a group theoretic characterization of twists inside Γ(S). It is rarely stated independently or emphasized but we would like to stress it in view of the profinite or procongruence case; we also refer to [25] for a nice proof based on the notion of stable rank. This is because first we do not know how to prove the profinite or procongruence analog, which is unfortunate, and second because in the profinite setting this would feature a rather striking and precise analog of the so-called “local correspondence” in birational anabelian geometry (see [21] for more detail). Armed with Theorem 2.4 it is easy to use Theorem 2.1 in order to study the automorphisms of Γ(S). Actually it turns out to be no more difficult to study morphisms between all the cofinite subgroups, (cf. [16], Theorem 2); we state this as:
Corollary 2.5. Assume d(S) > 1 and Γ = Γ(S) has trivial center; let Γ1, Γ2 ⊂ Γ be two finite index
subgroups. Then any isomorphism φ between Γ1 and Γ2 is induced by an element of M od(S), namely there
exists g ∈ M od(S) such that φ(g1) = g−1g1g for any g1∈ Γ1. In particular Out(Γ(S)) ' Z/2.
As usual one can study the two cases with nontrivial center, that is (1, 2) and (2, 0) in detail; see [25] for the latter one. This ends our review of the rigidity properties of the curves complex in the discrete setting, together with the group theoretic consequences. Before switching to the pants complex (or graph), we now introduce a kind of reconstruction technique for the various complexes.
2.2. Reconstructing complexes and the rigidity of the pants graph. In this paragraph we explore the local structure of our three complexes C(S), C∗(S) and CP(S) and show how to reconstruct them from
local data. We especially focus on the three graphs obtained by retaining only the 1-skeleta of C(S) and CP(S). As mentioned already we often abuse notation by writing CP(S) for the pants graph, bearing in mind
that the full two-dimensional complex can be reconstructed from its 1-skeleton (cf. [24]). Trivially we have CP(S) ,→ C∗(S); it will turn out that this inclusion or rather its (equally trivial) analog after completion is
of fundamental importance and lies in some sense at the very basis of a topological version of Grothendieck-Teichm¨uller theory. Note that (for d(S) > 1) C∗(S) is the 1-skeleton of the dual of the simplicial complex
C(S). In terms of automorphisms C∗(S) carries essentially the same information as C(S) (see below for a
precise statement) and it has been introduced essentially with a view to the above inclusion. Here we show (in the discrete setting) how to reconstruct the complexes from local data. Rigidity of the pants graph and a fortiori of the full complex will appear as an easy corollary. The proof of the reconstruction result (Theorem 2.10) is given in the next subsection.
Let us move to concrete and elementary notions. Given a surface S, a subsurface T is defined as T = S \ σ where σ ∈ C(S). We denote it Sσ; it is nothing but S cut or slit along the multicurve representing σ. In this
definition the curves are defined as usual up to isotopy and one can choose a representative of the multicurve. One way to do this in a coherent way is to equip S with a (any) metric of constant negative curvature and use the (unique) geodesic representatives of the various multicurves. The metric plainly induces a metric with the same property on all the subsurfaces of S. There is a natural inclusion C∗(Sσ) ⊂ C∗(S); in fact
C∗(Sσ) is the full subgraph of C∗(S) whose vertices correspond to those pants decompositions of S which
include σ (ditto for CP(S)). For σ ∈ C(S), we let |σ| denote the number of curves which constitute σ. So
|σ| = dim(σ) + 1 if dim(σ) denotes the dimension of the simplex σ ∈ C(S). The quantity |σ| turns out to be more convenient in our context; in particular d(Sσ) = d(S) − |σ|. We include throughout the case of an
empty cell (dimension −1): S∅= S. For example if σ is a maximal multicurve (pants decomposition), Sσ is
a disjoint union of pants and C∗(Sσ) is empty or reduced to a point (cf. §A.8) depending on convention. We
call two simplices ρ, σ ∈ C(S) compatible if the curves which compose ρ and σ do not intersect properly, that is they are either disjoint or coincide. Complex theoretically it means that ρ and σ lie in the closure of a common top dimensional simplex of C(S). If ρ and σ are compatible, we define their unions and intersections ρ ∪ σ, ρ ∩ σ ∈ C(S) in the obvious way. Then we clearly have:
Lemma 2.6. If ρ, σ ∈ C(S) are compatible simplices: C∗(Sρ) ∩ C∗(Sσ) = C∗(Sρ∪σ). If they are not
compatible, this intersection is empty.
Here all graphs C∗(Sτ) (τ ∈ C(S)) are considered as subgraphs of C∗(S). This lemma has a number of
equally obvious consequences. For instance C∗(Sρ) ⊂ C∗(Sσ) if and only if σ ⊂ ρ. Let us now return to the
connections between C∗ and CP. The inclusion CP ⊂ C∗ can be made more precise (cf. §A.8), given that
two simplicial embeddings of F in CP(S) are either disjoint, or else intersect in a single vertex.
Lemma 2.7. C∗(S) is obtained from CP(S) by replacing every maximal copy of the Farey graph F =
CP(S0,4) = CP(S1,1) inside CP(S) by a copy of the complete graph G = C∗(S0,4) = C∗(S1,1) associated to
the vertices of the given Farey graph.
A maximal copy of F is a subgraph of CP(S) which is isomorphic to F and is not properly contained
in another such subgraph. Note that the operation described in this lemma is not reversible; one cannot recognize CP(S) inside C∗(S) without additional information and this may well be the seed of
Grothendieck-Teichm¨uller theory. For the time being we note the following consequence in terms of automorphisms: Lemma 2.8.
Aut(CP(S)) ⊂ Aut(C∗(S))
Proof. An automorphism of CP(S) determines a permutation of the common vertex set V (S) (cf. §A.9),
which in turn defines an automorphism of C∗(S) provided it is compatible with its edges. Lemma 2.7 and
So any automorphism of CP(S) determines an automorphism of C∗(S) because both graphs share the
same set of vertices and automorphisms of complexes are determined by their effect on the vertices. However a priori only certain automorphisms of C∗(S) will preserve the additional structure given by the edges of
CP(S), inducing an automorphism of this subgraph. In dimension 1, Aut(G) is nothing but the permutation
group on its vertices. Any automorphism of F determines a unique automorphism of G by looking at its effect on the vertices, but Aut(F ) ' P GL2(Z) is certainly much smaller than Aut(G). In the discrete case
a kind of rigidification occurs for d(S) > 1 but this is not so after completion. Again this phenomenon lies at the very heart of Grothendieck-Teichm¨uller theory.
The (semi)local structure of C∗ and CP is not so mysterious. It is described in the following
Proposition 2.9. Let v ∈ V (S) be a vertex of C∗(S) and CP(S), with d(S) = k ≥ 0. Then v lies at the
intersection of exactly k maximal copies of G (resp. F ) in C∗(S) (resp. CP(S)). For any two copies Gi,
Gj (i 6= j) one has Gi∩ Gj = {v} ⊂ C∗(S) and two vertices wi∈ Gi, wj∈ Gj with wi6= v, wj6= v are not
joined by an edge in C∗(S).
As for F , for any two copies Fi and Fj (i 6= j) we have Fi∩ Fj= {v} ∈ CP(S) and for any wi∈ Fi such
that v and wi are connected by an edge, the vertices wi and wj are not connected by a finite chain in CP(S).
Proof. Let v be given as a pants decomposition v = (α1, . . . , αk). The main point here is that any triangle
(complete graph on three vertices) of C∗ or CP is obtained by varying one of the αi’s keeping all the other
curves αj fixed. This in turn depends only on the already mentioned (and obvious) fact that two curves on
a surface of dimension 1 always intersect. So we get k copies of G inside C∗which are indexed by the curves
appearing in v. The rest of the statement and the transposition to CP is easily verified.
Note that this shows that d(S) can be read off (graph theoretically) from C∗ or CP. In fact it can be
detected locally around any vertex v. To this end one can look for a star at v, namely a family (wi)i∈I of
vertices of C∗(S) such that each wi is connected to v by an edge and no two distinct wi’s are connected.
Then d(S) is the maximal possible number of such vertices i.e. the maximal cardinal of the index set I. Passing to CP(S), if wi, wj ∈ Fi ⊂ CP(S), then there is a finite chain connecting wi and wj in the link of
v. Together with the last assertion of the statement, this shows that there are exactly k = d(S) copies of F around v.
We now would like to reconstruct C(S) from C∗(S), hence also from CP(S) by Lemma 2.7. One way to do
this is to set up a correspondence between the subgraphs of C∗(S) which are graph theoretically isomorphic
to some C∗(Sσ) (σ ∈ C(S)) and the subsurfaces of S. This correspondence, to be later adapted to the
complete setting, is interesting even in this relatively simple discrete case. A precise wording goes as follows: Theorem 2.10. Let C ⊂ C∗(S) be a subgraph which is (abstractly) isomorphic to C∗(Σ) for a certain surface
Σ and is maximal with this property. Then there exists a unique σ ∈ C(S) such that C = C∗(Sσ).
The proof is deferred to the next subsection. Here we list some fairly straightforward and important consequences. First one has:
Corollary 2.11. C(S) can be (graph theoretically) reconstructed from C∗(S).
Proof. Starting from C∗(S) one builds a complex by considering subgraphs C as in the statement of the
theorem, with the inclusion map as boundary operator. The result ensures that this simplicial complex is
isomorphic to the curve complex C(S).
One then immediately gets:
Corollary 2.12. Aut(C∗(S)) = Aut(C(S)).
Taking Lemma 2.8 into account, this shows that there is a natural injective map: Aut(CP(S)) ,→ Aut(C(S)),
from which by Theorem 2.1 we get the rigidity of the pants graph (a fortiori the pants complex) as Theorem 2.13. Let S be a hyperbolic surface of type (g, n) with d(S) > 1. Then the natural map
θP : Inn(M od(S)) → Aut(CP(S))
is an isomorphism.
For the fact that here type (1, 2) is no exception, see the last page of [24], of which we thus reproved the main result. We will see below (in §7) how this rigidity result (Theorem 2.13) does survive (procongruence) completion, in sharp and interesting constrast with item i) of Theorem 2.1)
2.3. Proof of Theorem 2.10. Let us start with some remarks and reductions. First we note that the word “maximal” is indeed necessary. For instance there are proper subgraphs of F (resp. G) which are isomorphic to F (resp. G). Second, implicit in the statement is the fact that any C∗(Sσ) ⊂ C∗(S) does
indeed answer the problem, namely it is maximal in its isomorphy class. Assume on the contrary that we have a nested sequence C∗(Sσ) ⊂ C ⊂ C∗(S) where d(Sσ) = k, C is isomorphic to C∗(Sσ) and the first
inclusion is strict. Since C is connected, we can find a vertex w ∈ C \ C∗(Sσ) which is connected by an edge
to a vertex v ∈ C∗(Sσ). Since Sσ has dimension k, we can find k vertices wi ∈ C∗(Sσ) as in the proof of
Lemma 2.9 (with respect to v). But w ∈ C is connected to v and it is easy to check that it is not connected to any of the wi. In other words we have actually found k + 1 vertices which are connected to v and no two
of which are connected, which contradicts the fact that C is isomorphic to C∗(Sσ).
Having justified the statement, we can turn to the proof of Theorem 2.10, noticing first that uniqueness is clear: obviously C∗(Sσ) coincides with C∗(Sτ) (σ, τ ∈ C(S)) if and only if σ = τ ; this is also a very particular
case of Lemma 2.6. From Lemma 2.9 we can now define d(C) = d(Σ), which determines |σ| (assuming the existence of σ) since d(Sσ) = d(C) = d(S) − |σ|. Next the result is true if d(Σ) = 0 because then C∗(Σ) is
just a point and so is C. Hence it does correspond to a vertex of C∗(S), in other words to an actual pants
decomposition of S. We will prove the result by induction on k = d(Σ) but it is useful and enlightening to prove the case k = 1 directly. This is easy and essentially well-known in a different context. Much as in Lemma 2.9 the point is that any triangle inside C∗(S) (or CP(S)) determines a unique subsurface Σ with
d(Σ) = 1. This sets up a one-to-one correspondence between subsurfaces of S of dimension 1 and maximal complete subgraphs of C∗(S).
Now let k > 1, assume the result has been proved for d(C) < k and consider a graph C ⊂ C∗(S) as in the
statement, with d(C) = k. We fix an isomorphism C−∼→ C∗(Σ). Changing notation slightly for convenience,
we are looking for a subsurface T ⊂ S, defined by a cell of C(S) and such that C = C∗(T ). Note that it may
happen that the surfaces Σ and T (assuming the existence of the latter) are not of the same type because of the well-known exceptional low-dimensional isomorphisms between complexes of curves. One will have C∗(Σ) ' C∗(T ) and indeed, as a consequence of the result itself, C(Σ) ' C(T ), so for instance Σ could be
of type (0, 6) and T of type (2, 0).
We may now consider subsurfaces of Σ and transfer the information to C ⊂ C(S). Namely for any σ ∈ C∗(Σ), we denote by Cσ ⊂ C the subgraph corresponding to C∗(Σσ) under the fixed isomorphism
C ' C∗(Σ). Actually, forgetting about this isomorphism, we just write Cσ = C∗(Σσ) ⊂ C ⊂ C∗(S). By
the induction hypothesis, for any σ ∈ C(Σ), σ 6= ∅, there corresponds to Cσ a unique subsurface S(σ)∈ S.
Beware of the fact that σ now runs over the cells of C(Σ), not of C(S), and this is the reason of the added brackets. In these terms we are trying to extend this correspondence to σ = ∅, i.e. find T = S(∅).
In order to show the existence of T it is actually enough to show that there exists a k-dimensional subsurface of S, call it precisely T , such that any S(σ) with σ ∈ C(Σ) not empty is contained in T . Indeed,
the corresponding Cσ’s form a covering of C. So assuming the existence of such a subsurface T , we find that
C ⊂ C∗(T ); these two subgraphs being isomorphic and C being maximal by assumption, they coincide. In
order to prove the existence of T , we can now restrict attention to the largest possible S(σ)’s, i.e. to the case
|σ| = 1, which simply means that σ consists of a single loop.
We are thus reduced to showing that there exists a k-dimensional subsurface T ⊂ S such that, for any loop α on Σ, S(α) is contained in T . Now C(Σ) is connected because k > 1 and this can be used as
follows. If α and β are two non intersecting curves on Σ, Σα and Σβ are two subsurfaces of Σ of dimension
k − 1 intersecting along the subsurface Σα∪β of dimension k − 2, where α ∪ β is considered as a simplex of
C(Σ). Informally speaking for the time being, the union S(α)∪ S(β)has dimension k and this is the natural
candidate for T . In other words the latter, if it exists, is determined by any two non intersecting loops of Σ. Returning to the formal proof, let γ and δ be two arbitrary loops on Σ. There exists a path in the 1-skeleton of C(Σ) connecting γ to δ. It is given by a finite sequence γ, α1, . . . , αn, δ of loops such that α1
does not intersect γ, αn does not intersect δ and for 1 < i < n, αi does not intersect αi−1and αi+1. Using
the existence of such a chain, we are reduced to the following situation. Let α, β, and γ be three loops on Σ such that α ∩ β = β ∩ γ = ∅; there remains again to show that S(α), S(β) and S(γ) are contained in a
common k-dimensional subsurface T , and this will complete the proof of the result.
We can write S(α)= Sρ, S(β)= Sσ, S(γ) = Sτ, for certain simplices ρ, σ, τ ∈ C(S) with |ρ| = |σ| = |τ | =
d(S) − k + 1. Moreover, because α ∩ β = ∅ (resp. β ∩ γ = ∅) ρ and σ (resp. σ and τ ) are compatible simplices. So we can consider ρ ∩ σ and σ ∩ τ , with |ρ ∩ σ| = |σ ∩ τ | = d(S) − k. The corresponding surfaces Sρ∩σ and Sσ∩τ are both subsurfaces of S of dimension k. There remains only to show that they coincide:
is maximal in its isomorphy class. The complexes Cρ∩σ and Cσ∩τ are two subcomplexes of dimension k
inside C which is also of dimension k, and they are maximal such complexes, being attached to subsurfaces of S. This forces them to coincide – and in fact coincide with the whole of C. More formally, assume the contrary, that is Sρ∩σ and Sσ∩τ are distinct. Then, breaking the symmetry for a moment and relabeling if
necessary, we can choose as above two vertices v ∈ Cρ∩σ and w ∈ Cσ∩τ\ Cρ∩σ which are connected by an
edge. Then again pick a maximal family (wi) of k vertices in Cρ∩σ which are connected to v and are not
mutually connected. Adding in the vertex w we get a family of k + 1 vertices with the same properties,
which contradicts the fact that d(C) = k and completes the proof.
3. Profinite complexes and the isomorphism theorem
In this section we introduce and study profinite completions of the simplicial complexes which have ap-peared above. We focus on the procongruence completion because crucial results are not available to-date for the full profinite completions, as will become clear below. General foundations pertaining to completions of “spaces”, possibly equipped with group actions, are now available in a profinite context, thanks in particular to the work of G.Quick who has put these objects in the classical framework of model categories (see [31, 32] and references therein). However in our much more specific context we can and do rely on the more direct constructions of the first author (see [3, 4]). We then state and prove the crucial isomorphism result which very roughly speaking provides a bridge between group theoretic and complex or graph theoretic statements. We claim little novelty as to the framework and statements in this section, which are essentially borrowed from [4]. However some proofs in that paper (which itself uses [5] in a crucial way) are not so easy to decipher and it thus seemed useful to provide at times alternative proofs or at least sketches thereof, using a more concrete, if somewhat ad hoc approach. We have also added a short “guide for the perplexed” (§3.3) aiming at summarizing some of the main points of the theory, delineating a roadmap and pointing at a few serious bumps along the road.
3.1. Completions etc. Profinite complexes of curves were introduced in [3] ; the necessary constructions (and caveats) are summarized in [4], §3 to which we refer, especially concerning the congruence completions on which we focus hereafter. Minimal inputs appear in the Appendix below (§§A.10, 11). Starting as usual from a (connected) hyperbolic surface of finite type S and the attending Teichm¨uller group Γ = Γ(S), one constructs in particular its (full) profinite completion ˆΓ as well as its (pro)congruence completion ˇΓ (see §A.10). One then proceeds to show that the cofinite subgroups Γ(m)⊂ Γ (m > 2) pertaining to the abelian
levels M(m)(see again §A.10 or [4] for much more) operate without inversion on the (discrete) curve complex
C(S). This implies that this Γ-simplicial complex C(S) can be considered as a Γ(m)-simplicial set for any
m > 2 (after numbering the vertices). Now by restricting to the congruence levels which dominate some such abelian level (that is the inverse system of congruence subgroups Γλ with Γλ ⊂ Γ(m) ⊂ Γ for some
m > 2) we define the congruence completion ˇC(S) which we can view as a ˇΓ-simplicial profinite set, that is a simplicial object in the category of profinite sets, which moreover is equipped with an action of the congruence completion ˇΓ. We refer again to [3, 4] for the necessary precisions. Roughly speaking this makes sense of the definition of the congruence completion as a pro-simplicial set defined by
ˇ
C(S)•= lim←− λ∈Λ
C(S)•/Γλ
where Γλ runs over the congruence subgroups of Γ, indexed by the (countable) set Λ. We denote the finite
quotients by Cλ(S) = C(S)
•/Γλ. Note that one may and it is sometimes useful to restrict consideration to
the normal or even characteristic subgroups Γλ since both types define cofinal inverse subsystems (because
Γ is finitely generated). Note also that these completions are plainly defined “asymptotically”, that is one can omit any subsequence of “large” subgroups. This is why for instance we may restrict to congruence subgroups which are contained in some subgroup Γ(m) (m > 2).
So we regard ˇC(S)• as a simplicial object in the category of profinite sets, although below bullets are
often omitted, while keeping in mind that we are indeed dealing with simplicial objects. There is a canonical inclusion C(S) ,→ ˇC(S) ([4], Prop. 3.3) with dense image and a natural continuous action of ˇΓ on ˇC(S).
In a similar fashion and for the same reasons we can define ˇCP(S) as the inverse limit
ˇ
CP(S)•= lim←− λ∈Λ
CP(S)•/Γλ
and regard it again as a simplicial object in the category of profinite sets. It is in fact a prograph; the finite quotients are denoted Cλ
P(S) = C(S)•/Γλ. There is again a canonical inclusion CP(S) ,→ ˇCP(S) with dense
Γ(S) ,→ ˇΓ(S). Finally, as in the discrete case, there is a one-to-one correspondence between the vertices of ˇCP(S) and the simplices of ˇC(S) of maximal dimension (= d(S) − 1). A deep additional information is
contained in the edges of ˇCP(S).
We now concentrate on alternative, more geometric and manageable descriptions of the congruence curves complex ˇC = ˇC(S). More precisely we will shortly define the simplicial profinite complexes ˇCL= ˇCL(S) and
ˇ
CG = ˇCG(S), denoted respectively L(ˆπ) and L0(ˆπ) in [4] (ˆπ = ˆπ1top(S), the topological fundamental group
of the surface S) to which we refer for more detail. An important result, stated and proved in the next subsection asserts that ˇC(S), ˇCL(S) and ˇCG(S) are isomorphic, so that we are indeed describing the same
object from several standpoints. Typically, these three objects can be defined in the full profinite setting but the fact that they are isomorphic is not known.
In order to define ˇCL(S), where L stands for “loops” we first define its set of vertices ˆL(S) = ˇCL(S)0, the
set of unoriented proloops. Recall that in the discrete setting L(S) = CL(S)0 denotes the set of unoriented
simple loops up to isotopy which moreover are not peripheral, that is do not bound a disc on S with a single puncture. We are looking for a completion which is a priori simpler and more manageable than the one afforded by ˆC(S) in that it will involve only the fundamental group π = π1top(S) and its completion, instead of the much more involved Γ = π1top(M(S)), the topological or orbifold fundamental group of the moduli space of curves.
We proceed as follows (see again [4], §3). For a set X, let P(X) denote the set of unordered pairs of elements of X and for G a group, let G/ ∼ denote the set of conjugacy classes in G. Now given γ ∈ π, denote by γ± the equivalence class of the pair (γ, γ−1) in P(π) and by [γ±] the equivalence class of γ± in P(π/ ∼). Note that the latter has a natural structure of profinite set. The point is that there is a natural embedding ι : L ,→ P(π/ ∼). Indeed, given a loop ` ∈ L, it can be represented by an element γ = γ(`) ∈ π and we define ι(`) = [γ±], which is plainly independent of the choice of the representative γ of `. Finally
we define the set ˆL = ˆL(S) of proloops on S as the closure of the image ι(L) inside P(ˆπ/ ∼), where we are using the nontrivial fact from combinatorial group theory (conjugacy separability for the group π) that the natural map P(π/ ∼) → P(ˆπ/ ∼) is injective.
It is then easy to define, in much the same way, the simplicial complex CL(S) (with L = CL(S)0) and
its completion ˇCL(S) (with ˆL = ˇCL(S)0)). For X a set and k ≥ 1, define Pk(X) to be the set of unordered
subsets of P(X) with k elements (P = P1). Then we get a natural embedding ιk : C(S)k ,→ Pk+1(π/ ∼)
(ι = ι0) of the k-simplices of the discrete curve complex into the unordered sets of k + 1 conjugacy classes
of the group π modulo inversion. There remains only to define ˇCL(S)k as the closure of the image and to
organize the collection of profinite sets ( ˇCL(S)k) (0 ≤ k ≤ d(S) − 1) into the simplicial complex CL(S)•,
using the usual face and degeneracy operators (deleting and adding elements).
The last avatar ˇCG(S) of the congruence complex is actually easier to define. It is enough to define its
sets of vertices ˇCG(S)0 and then proceed as above. Return to L(S); mapping a simple loop to the cyclic
subgroup of Γ generated by the corresponding twist, we get a natural embedding L ,→ G(π)/ ∼ where the right-hand side denotes the set of cyclic subgroups of π modulo conjugacy. Again we have a further natural injective map G(π/ ∼) ,→ G(ˆπ)/ ∼ and we denote by ˆG(S) the closure of the image of L in G(ˆπ)/ ∼ via the composite embedding. Equivalently we may consider the image G(S) of L in G(π)/ ∼ and then take its closure ˆG(S in G(ˆπ)/ ∼, corresponding to certain procyclic subgroups of ˆπ, still up to conjugacy. Starting from ˆG(S) = ˆCG(S)0 we then build up the prosimplicial complex ˇCG(S) the same way we built ˇCL(S) out
of ˆL(S).
The next subsection will be essentially devoted to showing that these three avatars of ˇC(S) (including ˇ
C(S) itself) are isomorphic. Here in closing we add a few simple but extremely useful remarks about this type of relatively new objects. First of all one should keep in mind that we are dealing with compact (totally disconnected) spaces. This means in particular that there is no “going to infinity”. As a first extremely crude approximation ˆC(S) or ˇC(S) differ as much from C(S) as the ring Zpof the p-adic integrs differs from
Z. Note for instance that Thurston’s theory precisely starts from considerations connected with geometric intersection numbers, twists and ways of going to infinity, whether on Teichm¨uller space T (S) or on the curves complex C(S). Nothing of the kind is available – nor even relevant – here. For much more on a dynamical viewpoint on these objects we refer to [21], §8.
Next we sketch a line of arguments which we will meet below more than once. Let X(= X•) denote a
discrete G-simplicial complex with G a finitely generated group. Assume the number of G-orbits in X is finite. Let G0 be some completion of G and assume we have constructed a completion X0 of X which is a G0-prosimplicial complex. In particular there is a natural morphism ι : X → X0 with dense image and X0
through a unique φ0: X0→ Z i.e. φ = φ0◦ ι. Moreover ι is equivariant for the G-action on X and G0-action
on X0. Note that all the morphisms we consider are continuous for the natural topologies on their respective source and target.
In the situation above, there is at first a seemingly simple description of X0 which goes as follows. Pick k ≥ 0 and let Xk denote the k-skeleton of X ; by assumption one can decompose Xk into disjoint G-orbits
enumerated by the finite set Ek:
Xk =
a
σ∈Ek
G · σ,
where the k-simplices σ are representatives in the orbits. Under these circumstances one can decompose the k-skeleton Xk0 of X0 as
Xk0 = a
σ∈Ek
G0· ι(σ).
In other words it is covered by the G0-orbits of the images of the same simplices. Note that these orbits now may not be disjoint. In all the cases we will encounter X is residually finite, that is ι is injective, and we omit it from the notation. So X0 is made of (not necessarily disjoint) G0-orbits and there are finitely many in every dimension. The one line proof of the above is both simple and instructive. Consider the right-hand side of the equality above: it is compact because so is G0 and Ek is finite; it is dense because it contains
ι(X). So it coincides with Xk0.
Finally let X and Y be as above and for simplicity assume they are both residually finite so that we identify X (resp. Y ) with its image in X0 (resp. Y0). Let f : X → Y be a simplicial morphism. It naturally determines a morphism f0 : X0 → Y0 by the universality property of the completion. Moreover, if f is onto,
so is f0. The proof is again one line : the image f0(X0) contains f0(X) = f (X) = Y ⊂ Y0 which is dense ; f (X0) being dense and compact (as the continuous image of a compact) in Y0, it coincides with it.
3.2. The isomorphism theorem. Let us return to S and the three attending versions of the congruence complex, namely ˇC(S), ˇCL(S) and ˇCG(S). By the universality of the ˇΓ-completion we have a sequence of
(simplicial) maps :
ˇ
C(S) → ˇCL(S) → ˇCG(S).
We can now apply (twice) the reasoning immediately above (end of §3.1) and conclude that both maps are surjective. Their injectivity constitutes one of the main statements in [4] :
Theorem 3.1 ([4], Theorem. 4.2). The natural maps ˇC(S) → ˇCL(S) and ˇCL(S) → ˇCG(S) are ˇΓ-equivariant
isomorphisms of prosimplicial sets.
Sketch of proof. We will present a partial proof of this important result (using ideas from [4]), breaking it into three propositions. First it is clearly enough to show that the composition of the two maps is injective and one can actually restrict to showing that the map on the vertices, namely
Φ : ˇC(S)0→ ˇCG(S)0,
is injective, hence a bijection since it is known to be surjective. Recall that on the left ˇ
C(S)0= ˇL = lim←− λ∈Λ
L/Γλ
where λ ∈ Λ runs over the congruence subgroups of Γ. The right-hand side is given as the closure of the set of cyclic subgroups of π corresponding to elements of L(S) inside G(ˆπ)/ ∼, the set of procyclic subgroups of ˆπ (π = π1top(S)) modulo conjugacy. Both sides are naturally equipped with a ˇΓ-action and the map Φ is equivariant and onto. The only moot point is injectivity, whose validity is equivalent to that of the statement of the theorem. We used the symbol ˇL(S) because ˆL(S) has already been used for the set ˇCL(S)0of vertices
of ˇCL(S); a posteriori the theorem will confirm that ˇL(S) = ˆL(S).
Our first assertion reads:
Proposition 3.2. The map Φ induces a bijection between the respective ˇΓ-orbits of ˇC(S)0 and ˇCG(S)0.
In fact these ˇΓ-orbit have nothing mysterious. Indeed recall how curves and (Dehn) twists are related with the Γ-action in the discrete case. If α ∈ L is a loop (i.e. an isotopy class of simple closed curves), τα
the associated twist (we assume that the surface S has been given an orientation once and for all) and g ∈ Γ, then we have the familiar and elementary formula
Anticipating (a lot) we remark that the Grothendieck-Teichm¨uller action can and will be seen essentially as a generalization of this formula to ‘procurves’ and ‘protwists’. For the moment we recall that this provides a description of the Γ-orbits of the discrete complex C(S) = CL(S) = CG(S) (with obvious definitions). Two
loops α and β lie in the same Γ-orbit if and only if the topological types of the two slit surfaces Sα= S \ α
and Sβ= S \ β coincide. This is also the necessary and sufficient condition for the two associated twists τα
and τβ over α and β to be Γ-conjugate. The topological type of a twist τγ along a curve γ is defined as the
type of Sγ, the surface S slit along γ, which we also refer to as the type of the curve γ itself.
Now any ˇΓ-orbit in ˇC contains a discrete representative, i.e. a curve in L (see the end of §3.1). So the ˇ
Γ-orbits of ˇC are enumerated, with possible redundancies, by the finitely many topological types of the slit surfaces Sα(α ∈ L), which also enumerate the irreducible components of the divisor at infinity of the stable
compactification of M(S). The same is true of the ˇΓ-orbits of ˇCG, for the same reason. Since Φ is onto, this
shows that Proposition 3.2 is a consequence of the following:
Proposition 3.3. Given twists τα, τβ∈ Γ ⊂ ˇΓ (with α, β ∈ L) two nontrivial powers ταk, τβ` (k, ` ∈ ˆZ \ {0})
are conjugate in ˇΓ if and only if k = ` and τα and τβ (equivalently α and β) have the same topological type.
Note that this will show that the topological type of a “protwist”, or actually a power thereof is well-defined as the type of any discrete twist lying in the same ˇΓ-orbit, a protwist being nothing but a ˇΓ-conjugate of some bona fide discrete twist. We will henceforth often skip the prefix “pro” (“protwists”, “procurves”, etc.) when it should not lead to confusion. We also remark that it has long been known that the congruence levels separate the powers of a twist (or protwist for that matter). That is, given a twist τ , the natural map ˆ
Z →Γ which sends a ∈ ˆˇ Z to τa is injective. In other words the procyclic group hταi generated by a twist τα
is contained in ˇΓ.
Granted Proposition 3.3 (see below for its proof) there remains to show what we state as:
Proposition 3.4. For every α ∈ L = C(S)0⊂ ˇC(S)0 the ˇΓ-stabilizer ˇΓα⊂ ˇΓ of α as an element of ˇC(S)0
coincides with the stabilizer of its image Φ(α) ∈ ˇCG(S)0.
Here again one can reduce – as we did – the question to the stabilizer of a discrete curve by first acting with ˇΓ. Moreover, by [3], Proposition 6.5, the ˇΓ-stabilizer of α is the closure in ˇΓ of the stabilizer Γα ⊂ Γ
of α ∈ C(S)0, that is viewed as an element of the discrete complex C(S). Finally the discrete stabilizer Γα
affords an elementary geometric description.
We have now reduced the proof of Theorem 3.1 to those of Propositions 3.3 and 3.4. We will present the first in detail, partly for its own sake, partly in order to illustrate certain techniques in a concrete, if somewhat ad hoc way. By contrast, we will essentially rely on [4] for the proof of Proposition 3.4.
Proof of Proposition 3.3. First let us clarify the (natural) definition of a profinite power. If – say – g ∈ ˇΓ and k ∈ ˆZ, then gk ∈ ˇΓ is defined explicitly as an inverse system. For a level λ ∈ Λ, let a
λdenote the order
of the finite group Γ/Γλ. Then the λ-component of gk reads g
λkλ where gλ∈ Γ/Γλis the λ-component of g
and kλ∈ Z/aλ is the aλ-component of k (of course this definition is valid for any completion of any group).
Consider again ταand τβ, where α, β ∈ L(S). We need only prove the only if part of the statement: given
two profinite powers τk
α and τβ` (k, ` ∈ ˆZ \ {0}) there should exist a finite congruence quotient of Γ in which
their images are not conjugate, if either α and β do not share a common type, or k and ` are different. For α ∈ L(S), let Tαdenote the action in homology of τα. It is well-known that for any loop γ ∈ L(S) on
the surface we have
Tα[γ] = [γ] + h[γ], [α]i [α]
where [γ] denotes the homology class of the curve γ and h , i is the symplectic intersection form on S. Therefore Tαis either trivial, when the curve is separating (i.e. when [α] = 0), or it can be represented by
a nontrivial elementary matrix with one unit nonzero entry outside of the diagonal, when the curve α is nonseparating. Therefore the conjugacy classes of (powers of) twists along two curves, at least one of which is non separating, can be distinguished in any nontrivial congruence quotient of the integral symplectic group. We are thus reduced to the case where both α and β are separating, which we assume from now on.
Let f : ˜S → S be a characteristic (finite unramified) cover associated to a finite index characteristic subgroup K = π1( ˜S) ⊂ π = π1(S), which can be identified with the image f∗(π1( ˜S)) by the map f∗ induced
at the level of fundamental groups. It is Galois with Galois group GK = π/K. We would like to compute the
action in homology of the lift of a twist to such a cover. If φ ∈ Aut(π) is an automorphism of π, its restriction to K determines an automorphism ˜φ ∈ Aut(˜π)) (˜π = π1( ˜S) = K), via the requirement of equivariance
This also defines a way of lifting mapping classes ϕ ∈ Out+(π) = Γ = Γ(S) (where the superscript +
indicates the preservation of orientation) to ˜ϕ ∈ Out+(˜π) = ˜Γ = Γ( ˜S); the lift is well-defined up to the
action of the Galois group GK. Since mapping classes are determined by their action on the simple closed
curves we derive that ˜ϕ is determined – again up to multiplication by an element of GK – by the equivariance
on these, that is the property that
f ( ˜ϕ(˜γ)) = ϕ(f (˜γ)) (up to homotopy) for every γ ∈ L(S) and ˜γ ∈ L( ˜S) with γ = f∗(˜γ).
We say that a loop ˜α ∈ L( ˜S) on ˜S is a lift of α ∈ L(S) if it is a connected component of its preimage f−1(α); any two lifts of any two equivalent curves (i.e. curves with the same type) are equivalent. We denote by αn the curve obtained by traveling n times around α. If ˜α is a lift of α, the restriction of f to ˜α defines a finite covering of α of degree – say – m(α), which is independent of the choice of the lift, indeed only depends on the type (i.e. equivalence class) of α. In fact m(α) coincides with the order in GK = π/K of
any element of π represented by the curve α, which can be seen as follows. Let d(α) denote this order; it is well-defined since the various elements representing α belong to a single conjugacy class of GK. Then (the
class of) αd(α) belongs to K = f
∗(π1( ˜S)) and hence it can be lifted to a closed curve ˜α. Moreover ˜α cannot
be a power of some other curve ˜β ( ˜α = ˜βn, n > 1) because if so the restriction of f to ˜β would cover α with degree d(α)/n < d(α). Hence ˜α ∈ L( ˜S) and m(α) = d(α).
We wish to describe a lift ˜ταof τα to ˜Γ, or at least a suitable power of it. So let γ ∈ L(S), ˜γ ∈ L( ˜S) a lift
of γ. By the above
f (˜τα(˜γ)) = (τα(γ))d(γ).
On the other hand it also holds that
f (τα˜(˜γ)) = (ταd(α)(γ)) d(γ)
from which we conclude that
˜
ταd(α)= τα˜.
We can now compute the action of ˜ταd(α)on the homology of ˜S, which we denote ˜T d(α)
α . Indeed the action
of any power is determined by ˜
Tαkd(α)[˜γ] = Tα˜k[˜γ] = [˜γ] + kh[˜γ], [ ˜α]i[ ˜α]
with k an integer, ˜γ ∈ L( ˜S) and the angle brackets denote the symplectic pairing on ˜S (here and below pairings will implicitly relate to the relevant surface). From the above we find in particular that
˜
Tαkd(α)[˜γ] = [˜γ] + kh[˜γ], [ ˜α]i[ ˜α]. Raising this identity to the power d(β)) we find that
˜
Tαkd(α)d(β)[˜γ] = [˜γ] + kd(β)h[˜γ], [ ˜α]i[ ˜α]; swapping (α, k) and (β, `), this delivers
˜
Tβ`d(α)d(β)[˜γ] = [˜γ] + `d(α)h[˜γ], [ ˜β]i[ ˜β].
We now choose a basis of the integral homology group H1( ˜S) = H1( ˜S, Z) as follows. First consider the
curves ˜α and ˜β along with their images by the deck transformation group GK ; we then further adjoin simple
closed curves to this set until we reach a maximal set of pairwise disjoint curves which is invariant under the action of GK, so that no two curves are pairwise homotopic (and none is null homotopic).
Assume that the covering ˜S is such that the lifts ˜α and ˜β of both original curves α and β are nonseparating on ˜S. The validity of this crucial assumption will be discussed below. Granted this for the moment and using the basis of H1( ˜S) described above, the matrices corresponding to ˜T
kd(α)d(β) α , resp. ˜T `d(α)d(β) β read 1 Bα 0 1 resp. 1 Bβ 0 1
where Bα (resp. Bβ) is a diagonal matrix having d = |GK| nonzero entries equal to kd(α) (resp. `d(β)).
Note that GK acts via permutations on the basis of H1( ˜S).
Assume first that k = ` but α and β are not conjugate, and consider the principal congruence quotients of the integral symplectic group Aut(H1( ˜S), h , i) of the form Aut(H1( ˜S, Z/mZ), h , i) with m an integer.
We will show below that there exists an m such that the image of ˜Tαkd(α)d(β) is not conjugate to any
matrix in the GK-orbit of ˜T
kd(α)d(β)
β , which will imply that the images of ˜τ
kd(α)d(β) α = τ˜αk d(α)d(β) and ˜ τβkd(α)d(β) = ˜τk β d(α)d(β)
known that the latter is a congruence quotient of Γ, which will complete the proof of the proposition in that case. The other case, when k 6= ` but α and β are conjugate, is easy (one can assume that α = β).
In order to find a cover ˜S → S as described above, it is enough to find a characteristic subgroup K such that d(α) and d(β) are mutually prime. Indeed, picking then m = d(β) above, the image of ˜Tβkd(α)d(β)will be the identity modulo m along with all its GK-conjugates, whereas the image of ˜T
km(α)m(β)
α will be a nontrivial
unipotent, provided the curve ˜α is non separating on ˜S. Summarizing the above, in order to complete the proof of the proposition there remains to find a characteristic cover ˜S → S such that the lifts of α (and of β as well, in order to preserve symmetry) are nonseparating, whereas d(α) and d(β), that is the respective orders of the lifts of α and β in the group of the cover, are coprime; here recall that the lifts of a given loop, i.e. the connected components of its preimage, are conjugate in the group of the cover.
The two requirements above are essentially independent. First note that for any cover, the lifts of the nonseparating loops are nonseparating. Next it turns out to be easy to exhibit covers of S in which the lifts of all the separating loops, hence all the loops, are nonseparating (see below). Furthermore the lifts of the separating loops are simple : d(α) = 1 for any separating α ∈ L(S). (Note that here we are actually dealing with conjugacy classes of loops since the elements of L(S) are not attached to a base point, but that does not affect the argument.) Then any further cover has the property that all the lifts are nonseparating and there remains to manufacture such a cover with d(α) and d(β) coprime. We can thus break the remaining part of the proof of the proposition into two lemmas, the first of which reads:
Lemma 3.5. For any integer m ≥ 1, consider the cover S(m) corresponding to the invariant subgroup π(m) which is the kernel of the natural surjection
p(m): π = π1(S) → H1(S, Z/m).
Then the lifts of all the loops on S to S(m) are non separating and simple.
Proof. Let α ∈ L(S) be a loop on S. If α is non separating there is nothing to prove. If it is, then its image [α] in homology is trivial, and so in particular is its reduction in H1(S, Z/m). In other words π(m) contains
all the separating loops. So we find that d(α) = 1 which, referring to the above, implies that the multiplicity (= d(α)) of any connected component of the preimage p−1(m)(α) is also equal to 1. But this says that this preimage breaks into dm= |H1(S, Z/m)| non separating curves, whose union separates S(m).
Here are some additional remarks. First the covers S(m)are precisely those which are used when defining
the abelian levels M(S)(m) and the principal congruence subgroups Γ(S)(m)⊂ Γ(S) (see e.g. [5], §1). Then
π(m)= [π, π] · πmis a cofinite invariant subgroup of π and so is K(m)= [K, K] · Kmfor any cofinite invariant
subgroup K ⊂ π. Lemma 3.10 in [5] (whose proof is much trickier) asserts that all the covers associated to such subgroups (under some mild additional conditions) have the property that the inverse image of a loop does not contain separating loops. Indeed it states much more which we refrain from detailing here. This provides a much larger sample of covers and constitutes the basis for the essential “linearization” of the tower of congruence subgroups of Γ(S) (see §3.3 below). Finally and in a different vein, we note the tantalizing analogy between the breaking of the preimage of separating loops in a Galois cover and the completely split primes in a Galois field extension. We now turn to the second and concluding lemma namely:
Lemma 3.6. Given α, β ∈ L(S) there exists a finite, unramified, Galois cover of S such that d(α) and d(β) are coprime.
Proof. Fix two coprime numbers ` and m. Consider the quotient group Π = Π(`,m)= π1(S)/hα`= 1, βm= 1i
This is the fundamental group of a 2-complex obtained by adding 2-cells along the relations. It splits as an amalgamated product Π = Π1∗hαiΠ2∗hβiΠ3 where:
Π1= π1(S1)/hα`= 1i, Π2= π1(S2)/hα`= βm= 1i, Π3= π1(S3)/hβm= 1i.
Notice now that the Πj’s are fundamental groups of orbifolds, namely they are Fuchsian groups of nonzero
genus. In particular they are conjugacy separable, hence they admit finite quotients in which α has order ` and β has order m. Pick such finite quotients Qj of Πj, so that Π surjects onto an amalgamated product
Q = Q1∗hαiQ2∗hβiQ3. It is well-known that a graph of groups in which the vertex groups are finite, such
as Q, is virtually free. Let now Q denote a finite quotient of Q such that ker(Q → Q) is free. Then the images of α and β have respective orders ` and m in Q, since the kernel is torsionfree.
Consider next a finite index characteristic subgroup K of π contained in ker(π → Q), for instance the intersections of its images by all the conjugacy automorphisms. Then GK surjects onto Q and in particular
the orders of α and β in GK are divisors of ` and m, respectively. In particular, these are coprime integers.
This completes the proof of the lemma, hence also of Proposition 3.3.
As mentioned above we refer to [4] for the proof of Proposition 3.4, which will complete the proof of Theorem 3.1. In fact the core of the proof of Proposition 3.4, to be found at the very end of the proof of Theorem 4.2 in [4] (top of p.5200) consists in a direct application of the “linearization theorem” in [5], to which we return in the next subsection. In essence it does not differ so much from the proof of Proposition 3.3 presented above, which is in line with the proof of the linearization theorem.
Thanks to the isomorphism theorem we will henceforth often refer to the (pro)congruence curve complex, without explicitly distinguishing between its three versions, namely ˇC(S), ˇCL(S) and ˇCG(S). As a last item
in this paragraph we mention a fairly direct consequence of Proposition 3.3, namely:
Proposition 3.7. The ˇΓ(S)-orbits of the simplices of the procongruence complex ˇC(S) are in one-to-one correspondence with the Γ(S)-orbits of the simplices of the discrete complex C(S).
Proof. It is enough to show that if two discrete (k − 1)-simplices α = {α1, . . . , αk} and β = {β1, . . . , βk},
as viewed in ˇC(S), sit in the same ˇΓ-orbit, then they actually belong to the same Γ-orbit. Proposition 3.3 takes care of the case of loops (k = 1) and then one proceeds by induction. Assuming α and β are in the same ˇΓ-orbit, Proposition 3.3 says there exists g ∈ Γ such that g(α1) = β1. After twisting by g we may thus
assume that α1= β1. Now by assumption there exists h ∈ ˇΓ such that h(α) = β and h belongs to ˇΓα1, the
stabilizer of the loop α1in ˇΓ. By [4] (Theorem 4.5) this stabilizer is naturally isomorphic to an extension of
ˇ
Γ(Sα1) by the procyclic group hτα1i generated by the twist along α1. Here Sα1 denotes as usual the surface
S slit along the loop α1 and note that we are using the fact that we consider precisely the procongruence
completion (see [3], Proposition 6.6). Multiplying out by a (profinite) power of the twist τα1, we are led
to dealing with (k − 2)-simplices on the surface Sα1, where the assertion holds true by induction, which
proceeds either on the dimension of the simplices or on the modular dimension of the underlying surface S. 3.3. Elucidation. Before moving forward it may be desirable, indeed necessary, to elucidate the actual content of the above isomorphim result and its significance. The point is roughly that objects which are more or less clearly equivalent (isomorphic) in the discrete case, are definitely not obviously so after completion. Sometimes the equivalence requires a difficult proof and sometimes it simply does not hold true. So let us first briefly review the various objects connected with isotopy classes of simple closed curves (a.k.a. loops) on a connected oriented hyperbolic surface S. We will essentially confine ourselves to the case of a single loop, higher simplices are determined by their vertices.
Let us first summarize and review four constructions, starting from an oriented loop ~γ on S, where we may consider that ~γ ∈ π = π1(S). Since π is constructed picking out a basepoint P ∈ S this means that we
choose a loop through P in the free isotopy class of ~γ. Let γ ∈ L(S) = C(S)0denote ~γ after forgetting the
orientation.
Working again with the fundamental group π, specifying γ is equivalent to specifying a pair γ± of two oriented loops with opposite orientations. Passing to conjugacy classes in order to free the construction from the choice of a basepoint, we find that γ ∈ L leads to an unordered pair [γ±] of elements of π/ ∼ which is now an element CL(S)0.
That was so to speak on the graph theoretic side. Now from a group theoretic viewpoint, γ defines the cyclic subgroup hγi ⊂ π it generates inside π (~γ and ~γ−1 define the same subgroup). Considering the
subgroup hγi up to conjugacy in π leads to the definition of γ as an element of CG(S)0. Slightly more
generally, given any integer k > 0, one can consider the finite index cyclic subgroup hγki ⊂ π. This will
prove useful below.
Finally one can pass to the Teichm¨uller group Γ(S). Then γ defines the twist τγ along it (using the
orientation of S) and again the cyclic group hτγi ⊂ Γ(S) or its finite index subgroups hτγki ⊂ Γ(S) (k > 0).
So far so good in the discrete case. Part of the fundational work then consists in exploring what happens after completion. A main point is that one can complete either working directly with π, the fundamental group of the surface S, and thus its profinite completion ˆπ, or with Γ = Γ(S), the fundamental group of the moduli space M(S). It is clear a priori that these two forms of completions can be related only if one considers completions of Γ that are no finer than the congruence completion ˇΓ, which records the covers of
M(S) coming from covers of S. Recall that the congruence conjecture asserts that in fact ˇΓ = ˆΓ. So in some sense the problem, from this foundational standpoint, consists in setting up a dictionary between these two kinds of completions, and also, in a slightly different but closely related fashion, between the graph theoretic and the group theoretic information.
Concretely, what are then the main tools and results ? We will list one essential tool and two foundational results, globally referring to [4, 5]. Let us give these threee statements names as it can help further reference as well as pointing to the core of the matter. The tool leads to a kind of linearization of the problem, replacing homotopy with homology. The first result is precisely the isomorphism theorem above (Theorem 3.1) ; the second one expresses a property we will refer to as twist separability. Let us now go into somewhat more detail.
The idea of “linearization” is fairly old and may be ascribed to E.Looijenga. It has actually been used in the proof of Proposition 3.3 above. A general expression of this principle is embodied by Corollary 7.8 in [4]. A proper statement is cumbersome and requires introducing a lot of notation, so let us content ourselves with the main idea, namely that given S as above, a loop γ ∈ L(S) is entirely determined by the projective set of the homology classes of its preimages on the (finite unramified) covers of S. Explicitly and with π = π1(S),
let K ⊂ π an invariant finite index subgroup (normal would be enough but invariant is forced when working with mapping class groups), let GK = π/K denote the quotient group, pK : SK → S the ensuing Galois
cover with group GK. For α ∈ L(T ) a loop on a surface T , let [α] ∈ H1(T, Z) denote the associated integral
homology class. Then given γ ∈ L(S), we can consider the projective system ([p−1K (γ)])K of homology classes
on SK, where K runs through the cofinite invariant subgroups of π (for K = π, SK = S and we omit the
mention of pπ= id). Roughly speaking, the theorem asserts that γ is entirely determined by the family of
“linear” data ([p−1K (γ)])K.
What are the obvious obstacles which arise when trying to identify a loop via its homology class? In fact, a loop α ∈ L(S) is trivial in homology, that is [α] = 0 ∈ H1(S), if and only if α is separating. More generally,
given non intersecting loops α, β ∈ L(S), their homology classes coincide ([α] = [β]) for the appropriate orientations if and only if they form a cut pair, that is their union separates the surface (the first case can be seen as the case β = ∅). This is why it is important to detect a large sample of covers pK : SK → S such
that for any loop on S, more generally any simplex σ ∈ C(S), the inverse image p−1K (σ) does not contain separating curves nor cut pairs. This is provided by the important Lemma 3.10 in [5] (see above, after the proof of Proposition 3.3).
Passing to the first main result, it was already mentioned that it is embodied by the isomorphism theorem (Theorem 3.1) above. Here we simply insist again that its main thrust lies in connecting, on the one hand completion via the Teichm¨uller group Γ(S) i.e. the fundamental group of the moduli space M(S), which is used when defining ˇC(S), on the other hand completion via the much simpler and more tractable fundamental group π = π1(S) of the surface S itself, which is used when defining both ˇCL(S) and ˇCG(S).
The second main result traces a fundamental link between (pro)curves and (pro)twists, that is between the graph theoretic and the group theoretic facets of the theory. This is Theorem 5.1 in [4], which can be stated more easily. Starting in the discrete setting we have (after orienting the surface S) a natural injective map d : L(S) ,→ Γ(S) which to a loop γ ∈ L(S) assigns the corresponding twist τγ. Given k ∈ Z \ {0} it can
be generalized to dk : γ 7→ τγk (d = d1), still an injective map between the same source and target.
As usual, upon completion the plot thickens and things become more interesting. From the injective map d and the natural embedding Γ ,→ ˇΓ we get a (still injective) map which we denote by the same name for simplicity d : L ,→ ˇΓ. By the universality of the progruence completion, this leads to a map
ˆ
d : ˇL(S) → ˇΓ(S)
which now may or may not be injective (this is precisely the moot point here) with, as above, ˇ
L(S) = ˇC(S)0= lim←− λ∈Λ
L(S)/Γλ.
Finally the isomorphism theorem ensures that ˇL(S) = ˆL(S) = ˇCL(S)0, namely the set of (pro)curves on S.
This can be generalized in the obvious way to ˆdk : ˇL(S) → ˇΓ(S) for any k ∈ Z \ {0} and indeed jazzed up
to k ∈ ˆZ \ {0}, using the density of Z inZ. Note from a topological viewpoint that in the complete case weˆ are always considering continuous maps between compact spaces.
We may now state the second fundamental result about twists separability we have been alluding to: Theorem 3.8 ([4], Thm. 5.1). For any k ∈ ˆZ \ {0} the map
ˆ
